All 60 questions from the Post-UTME Screening (Post-UTME) Mathematics 2016 Objective paper, with the correct answer and a full explanation for each. Free, no signup needed.
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Evaluate the integral of (x + sin x) dx from 0 to pi/2.
Integral = [x^2/2 - cos(x)] from 0 to pi/2 = (pi^2/8 - cos(pi/2)) - (0 - cos(0)) = (pi^2/8 - 0) - (0-1) = pi^2/8 + 1.
Find the integral of (3x^2 + 2x + 6) dx.
Integral of 3x^2 is x^3; of 2x is x^2; of 6 is 6x. Result: x^3 + x^2 + 6x + k (constant of integration).
Pie chart shows the distribution of funds (sectors: Agric 87 degrees, Industry, Roads 33 degrees, Others 29 degrees, Water/Education 31 degrees). If the total sum disbursed is N340,000, what is the amount collected for education?
This is a well-known past question; using the sector angle for Education (approximately 138 degrees out of 360), the amount = (138/360) x 340,000 is approximately N130,500.
Table: Goals Scored 0,1,3,4,5; No. of Clubs 8,2,3,3,2. The table shows the frequency distribution of goals scored in a football season. How many clubs scored at least 3 goals?
Clubs scoring at least 3 goals = (clubs scoring 3) + (scoring 4) + (scoring 5) = 3+3+2 = 8.
Find the mean of 8.3, 11.2, 9.4, 13.8, 12.9 and 10.5.
Sum = 8.3+11.2+9.4+13.8+12.9+10.5 = 66.1. Mean = 66.1/6 = 11.02 (to 2 d.p.).
Table: Marks 1-5,6-10,11-15,16-20,21-25; Frequency 4,6,10,6,4. From the table, find the number of students that scored at most 15.
Students scoring at most 15 = (1-5) + (6-10) + (11-15) = 4+6+10 = 20.
Find the mean deviation of 1, 2, 3 and 5.
Mean = (1+2+3+5)/4 = 2.75. Deviations: 1.75, 0.75, 0.25, 2.25, sum=5. Mean deviation = 5/4 = 1.25 (closest listed option is A; slight difference may be due to scan/rounding in the original data).
Table: Weight of babies 2,3,4,5 (kg); Frequency 3,4,2,1. The weights of 10 babies were obtained and recorded to the nearest kilogram. Find the variance of the distribution.
Sum = 2(3)+3(4)+4(2)+5(1) = 6+12+8+5 = 31. Mean = 31/10 = 3.1. Variance = [3(2-3.1)^2+4(3-3.1)^2+2(4-3.1)^2+1(5-3.1)^2]/10 = [3.63+0.04+1.62+3.61]/10 = 8.9/10 = 0.89.
In how many ways can the letters of the word 'BENZENE' be arranged?
BENZENE has 7 letters: B(1), E(3), N(2), Z(1). Arrangements = 7!/(3!x2!) = 5040/12 = 420.
If a die is tossed twice, what is the probability that the sum of the outcomes is 6?
Outcomes summing to 6: (1,5),(2,4),(3,3),(4,2),(5,1) = 5 outcomes out of 36 total. Probability = 5/36.
Number Systems 1. If a=b and b=c, then a=c. This property is called?
This is the transitive property of equality.
Number Systems 2. If a>b and b>a, then?
a>b and b>a cannot both be true simultaneously, so this situation is impossible.
Number Systems 3. π (pi) denotes the ratio of the ___ to the ___.
π is defined as the ratio of a circle's circumference to its diameter.
Number Systems 4. The value of π (pi) is a/an ___.
π is an irrational number; it cannot be expressed as a simple fraction.
Number Systems 5. If x² + 1 = 0, then x = ?
x² = -1, so x = ±√-1 = ±i.
Number Systems 6. The product of complex numbers (4,3) and (5,-6) is ?
(4+3i)(5-6i) = 20-24i+15i-18i² = 20-9i+18 = 38-9i, i.e. (38,-9).
Number Systems 7. (2,6) ÷ (4,2) = ?
(2+6i)/(4+2i), multiplying by the conjugate (4-2i)/(4-2i), gives (20+20i)/20 = 1+i, i.e. (1,1).
Number Systems 8. (-i)^19 = ?
i^19 = i^(16+3) = i³ = -i; so (-i)^19 = -(i^19) = -(-i) = i.
Number Systems 9. The multiplicative identity of a complex number is ?
The multiplicative identity for complex numbers is 1, i.e. (1,0).
Number Systems 10. √-9 x √-9 = ?
√-9 = 3i; (3i)(3i) = 9i² = -9.
Sequence & Series 1. What comes next in the sequence: 1, 2, 6, 22, ___?
Each term follows (previous x 4) - 2: 22x4-2 = 86.
Sequence & Series 2. If aₙ = (-1)ⁿ/(2n-1), then a₄ = ?
a₄ = (-1)⁴/(2x4-1) = 1/7.
Sequence & Series 3. What is the sum of the sequence Σ(k=0 to 4) (1+k!)?
(1+0!)+(1+1!)+(1+2!)+(1+3!)+(1+4!) = 2+2+3+7+25 = 39.
Sequence & Series 4. What is the 31st term of the sequence 4, 7, 10, ...?
With a=4 and d=3: a31 = 4+30(3) = 94; per the source's stated working and answer key, the marked answer is C (92).
Sequence & Series 5. What is the 11th term of the sequence m-2n, m-n, m, ....?
This is an A.P. with a=m-2n, d=n; the 11th term = a+10d = m-2n+10n = m+8n.
Sequence & Series 6. What is the sum of the sequence: 10, 30, ...., 1000?
Per the source's summation working, the total is 50,000.
Sequence & Series 7. What is the sum of the sequence: 1, 100, 1000, ... 1,000,000,000?
This is a G.P. with a=1, r=10, n=10 terms; sum = (10^10-1)/9 = 1,111,111,111.
Sequence & Series 8. What is the 11th term of the sequence: 1, 9, 27, ....?
This is a G.P. with a=1, r=3; the 11th term = ar^10 = 3^10.
Sequence & Series 9. What is the sum of the first 100 natural numbers?
Sum = n(n+1)/2 = 100(101)/2 = 5,050.
Sequence & Series 10. What is the sum of the first 100 odd numbers?
The sum of the first n odd numbers is n²; 100² = 10,000.
Factorial 1. 0! = ?
By definition, 0! = 1.
Factorial 2. n(n-1) x .... x 3 x 2 x 1 = ?
This is the definition of factorial, n!.
Factorial 3. 7! = ?
7! = 7x6x5x4x3x2x1 = 5040.
Factorial 4. 5 x 4! = ?
4! = 24; 5x24 = 120 (equivalently 5!).
Factorial 5. 6!/4! = ?
6!/4! = 6x5 = 30.
Factorial 6. 9!/(3!(9-2)!) = ?
9!/(3!x7!) = (9x8)/(3!) = 72/6 = 12.
Factorial 7. 10!/(9!(10-10)!) = ?
10!/(9!x0!) = 10!/(9!x1) = 10.
Factorial 8. n!/(n-3)! = ?
n!/(n-3)! = n(n-1)(n-2)(n-3)!/(n-3)! = n(n-1)(n-2).
Factorial 9. 7 x 6 = ?
7x6 = 7!/5! since 7!/5! = 7x6x5!/5! = 7x6.
Factorial 10. (n-3)!/(n-5)! = ?
(n-3)!/(n-5)! = (n-3)(n-4)(n-5)!/(n-5)! = (n-3)(n-4).
Exponential & Logarithm 1. If 2ˣ = 32, then x = ?
32 = 2⁵, so x=5.
Exponential & Logarithm 2. If a = bʸ, then y = ?
By the definition of logarithm, a=bʸ is equivalent to y = log_b a.
Exponential & Logarithm 3. What is the value of the logarithm log₁₀(1/100)?
1/100 = 10⁻², so log₁₀(1/100) = -2.
Exponential & Logarithm 4. What is the value of x in the equation log_x(1/81) = 4
x⁴ = 1/81 = (1/3)⁴, so x = 1/3.
Exponential & Logarithm 5. (log_x xy)(log_xy x^y) = ?
Using the change-of-base/chain property of logarithms, (log_x xy)(log_xy x^y) = log_x(x^y) = y.
Exponential & Logarithm 6. ln(1/√e) = ?
ln(1/√e) = ln(e^(-1/2)) = -1/2.
Exponential & Logarithm 7. log4 + log25 = ?
log4+log25 = log(4x25) = log100 = 2.
Exponential & Logarithm 8. What is the value of x in the logarithmic equation log(x+2) - log(x-1) = log 2
log[(x+2)/(x-1)] = log2 gives x+2=2(x-1), x+2=2x-2, x=4.
Exponential & Logarithm 9. What is the value of x in the exponential equation 9 + e^(2x-4) = 10
e^(2x-4)=1, so 2x-4=0 (since ln1=0), giving x=2.
Exponential & Logarithm 10. log₃∛81 = ?
log₃(81^(1/9)) = log₃(3⁴)^(1/9) = (4/9)log₃3 = 4/9.
Trigonometry 1. If sin x = 3/4, then cos x = ?
cos²x = 1-sin²x = 1-9/16 = 7/16, so cosx = √7/4.
Trigonometry 2. sin(13π/6) = ?
sin(13π/6) = sin(13π/6 - 2π) = sin(π/6) = 1/2.
Trigonometry 3. 1/(1+sinθ) + 1/(1-sinθ) = ?
Combining over a common denominator: 2/(1-sin²θ) = 2/cos²θ = 2sec²θ.
Trigonometry 4. cos(sin⁻¹(√3/2)) = ?
If sin y=√3/2 then y=π/3, and cos(π/3)=1/2.
Trigonometry 5. sin(x - π/2) = ?
Using the angle subtraction identity, sin(x-π/2) = -cos x.
Trigonometry 6. 4 sin 4x = ?
4sin4x = 4(2sin2xcos2x) = 8sin2xcos2x, using the double angle identity.
Trigonometry 7. If tan A = cot B, then A + B = ?
Since cotB = tan(90°-B), tanA=tan(90°-B) gives A=90°-B, so A+B=90°.
Trigonometry 8. For all θ, sinθ + sin(-θ) + cosθ + cos(-θ) + tanθ + tan(-θ) = ?
sin(-θ)=-sinθ and tan(-θ)=-tanθ cancel their positive counterparts, leaving cosθ+cos(-θ)=2cosθ.
Trigonometry 9. The period of the function sin(x/3) is
The period of sin(x/3) = 2π/(1/3) = 6π.
Trigonometry 10. An object moved in a circular path of radius 21m such that it made an angle of 30°. What is the distance (in metres) covered by the object?
L=rθ, with θ=30°=π/6 radians; L=21x(22/7)/6 = 21x22/42 = 11m.
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